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MODELING OF THE SOLAR SOURCE

1.1 INTRODUCTION

Solar energy is the portion of the Sun's radiant heat and light, which is available at the Earth's surface for various applications of generating energy, that is, converting the energy form of the Sun into energy for useful applications. This is done, for example, by exciting electrons in a photovoltaic cell, supplying energy to natural processes like photosynthesis, or by heating objects. This energy is free, clean, and abundant in most places throughout the year and is important especially at the time of high fossil fuel costs and degradation of the atmosphere by the use of these fossil fuels. Solar energy is carried on the solar radiation, which consists of two parts: extraterrestrial solar radiation, which is above the atmosphere, and global solar radiation, which is at surface level below the atmosphere. The components of global solar radiation are usually measured by pyranometers, solarimeters, actinography, or pyrheliometers. These measuring devices are usually installed at selected sites in specific regions. Due to high cost of these devices, it is not feasible to install them at many sites. In addition, these measuring devices have notable tolerances and accuracy deficiencies, and consequently wrong/missing records may occur in a measured data set. Thus, there is a need for modeling of the solar source considering solar astronomy and geometry principles. Moreover, the measured solar radiation values can be used for developing solar radiation models that describe the mathematical relations between the solar radiation and the meteorological variables such as ambient temperature, humidity, and sunshine ratio. These models can be later be used to predict solar radiation at places where there is no solar energy measuring device installed.

1.2 MODELING OF THE SUN POSITION

As a fact, the Earth rotates around the Sun in an elliptical orbit. Figure 1.1 shows the Earth rotation orbit around the Sun. The length of each rotation the Earth makes around the Sun is about 8766 h, which approximately stands for 365.242 days.

FIGURE 1.1 Earth rotation orbit around the Sun.

From the figure, it can be seen that there are some unique points at this orbit. The winter solstice occurs on December 21, at which the Earth is about 147 million km away from the Sun. On the other hand, at the summer solstice, which occurs on June 21, the Earth is about 152 million km from the Sun. However, to provide more accurate points, the Earth is closest to the Sun (147 million km) on January 2, and this point is called perihelion. The point where the Earth is furthest from the Sun (152 million km) is called aphelion and occurs on July 3.

For an observer standing at specific point on the Earth, the Sun position can be determined by two main angles, namely, altitude angle (a) and azimuth angle (?S), as seen in Figure 1.2.

FIGURE 1.2 The Sun's altitude and azimuth angles.

From Figure 1.2 the altitude angle is the angular height of the Sun in the sky measured from the horizontal. The altitude angle can be given by

where L is the latitude of the location, d is the angle of declination, and ? is the hour angle.

The angle of declination is the angle between the Earth-Sun vector and the equatorial plane (see Fig. 1.3) and is calculated as follows (results in degree, arguments to trigonomic functions are expected to be in radiant):?S

FIGURE 1.3 Solar declination angle.

The hour angle, ?, is the angular displacement of the Sun from the local point, and it is given by

where AST is apparent or true solar time and is given by the daily apparent motion of the true or observed Sun. AST is based on the apparent solar day, which is the interval between two successive returns of the Sun to the local meridian. Apparent solar time is given by

where LMT is the local meridian time, LOD is the longitude, LSMT is the local standard meridian time, and EoT is the equation of time.

The LSMT is a reference meridian used for a particular time zone and is similar to the prime meridian, used for Greenwich Mean Time. LSMT is given by

The EoT is the difference between apparent and mean solar times, both taken at a given longitude at the same real instant of time. EoT is given by

where B can be calculated by

where N is the day number defined as the number of days elapsed in a given year up to a particular date (e.g., the 2nd of February corresponds to 33).

On the other hand, the azimuth angle as can be seen in Figure 1.2 is an angular displacement of the Sun reference line from the source axis. The azimuth angle can be calculated by

Example 1.1:

Develop a program in MATLAB® that calculates the altitude and azimuth angles at 13 : 12 on July 2, for the city of Kuala Lumpur.

Solution

The main parts of the program's structure are described as follows:

• Insert location coordinates (latitude and longitude), day number, and local mean time.
• Calculate angle of declination, equation of time, and LMST.
• Calculate AST and hour angle.
• Calculate altitude angle.
• Calculate azimuth angle.
• Plot results.

ANS: Alpha = 70.04°; theta = -1.13°

Example 1.2:

Modify the developed MATLAB code in Example 1.1 to calculate the altitude and azimuth angle profile (every 5 min) for the whole solar day of the 2nd of July for the city of Kuala Lumpur.

Solution

The solar day is defined as the duration from sunrise to sunset. Thus, the altitude and azimuth angles are required to be calculated for each hour from sunrise to sunset. The sunrise and sunset hour angles can be considered equal and calculated as

In the meanwhile, the solar time of each hour angle can be calculated by rewriting Equations 1.3 as follows:

The sign of Equation 1.10 must be minus if we want to calculate the sunrise time, while it must be plus if we are calculating the sunset time. Following that the main parts of the program's structure can be described as follows:

• Insert location coordinates (latitude and longitude) and day number.
• Calculate angle of declination.
• Calculate sunrise and sunset hour angles.
• Calculate AST of the sunrise and sunset.
• Calculate equation of time and LMST.
• Calculate the actual sunrise and sunset times.
• Set for a loop starting from the sunrise and terminating by the sunset with a step size of 5 min.
• Calculate the solar time and hour angle at each step.
• Calculate altitude angle at each step.
• Calculate azimuth angle at each step.
• Store the calculated altitude and azimuth angles in arrays.
• Plot the results.

ANS

FIGURE 1.4 A day's profile of the Sun's altitude and azimuth angles (Example 1.2).

1.3 MODELING OF EXTRATERRESTRIAL SOLAR RADIATION

The first step in modeling the solar source is to estimate the emitted radiation from the Sun. As a fact, the radiant energy of any emitting object can be described as a function of its temperature. The usual practice to estimate the radiant energy by an object is to compare it to a blackbody. A blackbody is defined as a perfect emitter and absorber. A perfect absorber can absorb all of the received energy with any reflections, while a perfect emitter emits energy more than any other object. Planck's law describes the wavelengths emitted by a blackbody at a specific temperature as follows:

where E? is the total emissive per unit area of blackbody emission rate (W/m2 µm), T is the absolute temperature of the blackbody (K), and ? is the wavelength (µm).

Example 1.3:

Develop a MATLAB code that calculates the spectral emissive power of a 288 K blackbody, for wavelengths in the range of (1-60) µm. After that calculate the power emitted between the wavelength of 20 and 30 µm.

Solution

The first part of the example can be solved by simply implementing Equation 1.11 and calculating its value for the requested wavelength range as follows:

FIGURE 1.5 Spectral emissive power of a 288 K blackbody, for wavelengths in the range of (1-60) µm (Example 1.3).

In order to calculate the emitted power between the wavelength value of 20 and 30 µm, the shaded area in Figure 1.5 can be calculated as follows:

which can be implemented in MATLAB as follows:

ANS = 704.0801 W/m2

The interior of the Sun is estimated to have a temperature of around...